Electrically controlled superconductor-to-failed insulator transition and giant anomalous Hall effect in kagome metal CsV3Sb5 nanoflakes

The electronic correlations (e.g. unconventional superconductivity (SC), chiral charge order and nematic order) and giant anomalous Hall effect (AHE) in topological kagome metals AV3Sb5 (A = K, Rb, and Cs) have attracted great interest. Electrical control of those correlated electronic states and AHE allows us to resolve their own nature and origin and to discover new quantum phenomena. Here, we show that electrically controlled proton intercalation has significant impacts on striking quantum phenomena in CsV3Sb5 nanodevices mainly through inducing disorders in thinner nanoflakes and carrier density modulation in thicker ones. Specifically, in disordered thin nanoflakes (below 25 nm), we achieve a quantum phase transition from a superconductor to a “failed insulator” with a large saturated sheet resistance for T → 0 K. Meanwhile, the carrier density modulation in thicker nanoflakes shifts the Fermi level across the charge density wave (CDW) gap and gives rise to an extrinsic-intrinsic transition of AHE. With the first-principles calculations, the extrinsic skew scattering of holes in the nearly flat bands with finite Berry curvature by multiple impurities would account for the giant AHE. Our work uncovers a distinct disorder-driven bosonic superconductor-insulator transition (SIT), outlines a global picture of the giant AHE and reveals its correlation with the unconventional CDW in the AV3Sb5 family.


Supplementary section 2: Transport properties of CsV 3 Sb 5 crystals
CsV3Sb5 single crystal exhibits CDW transition at 90 K, the superconducting transition at 2.5 K and Shubnikov-De Hass (SdH) oscillations under high fields at 2 K, as shown in Supplementary Fig. 3a and 3b. Anomalous Hall effect within ±2 T was also observed in bulk CsV3Sb5 crystals. Supplementary Fig. 3c and 3d show the Hall resistivities of bulk CsV3Sb5 at various temperatures and the anomalous Hall resistivities (AHE) extracted by subtracting the linear Hall background at high fields, respectively. The anomalous Hall conductivity (AHC) was obtained by inverting the resistivity matrix, = − ( 2 + 2 ) ⁄ , as shown in the inset of Supplementary Fig. 2d. Recent spectroscopy experiments revealed that the Fermi level of pristine CsV3Sb5 lies slightly above (van Hove singularity) VHS1 near the M points [1]. The observed superconductivity, CDW as well as giant AHE in our bulk CsV3Sb5 are qualitatively in line with those reported single crystals [2]. These similarities revealed that our crystal should share the same Fermi level with those reported high quality CsV3Sb5 crystals. In our exfoliated nanoflakes, however, the Fermi level would be shifted slightly downward, due to the Cs vacancies [3][4][5], as discussed in the main text. The inset shows the AHC components after inverting the resistivity matrix.
Magnetic field (T)   [16]). First, in ferromagnetic (FM) metals, the spin-orbit coupling leads to the asymmetric scattering of a carrier from impurities and introduces a momentum perpendicular to both incident momentum and the magnetization. This scenario had been detailedly examined in 2D FM Rashba model [17] and 2D massive Dirac fermions [18] with nonzero Berry curvature and finite AHE. Both the scalar and magnetic impurities could induce the extrinsic AHE including the skew scattering and side jump contributions. The linear scaling law between the anomalous Hall conductivity and the longitudinal one was derived [17]. Note that the finite temperature could enhance the scattering rate of carriers against phonons and decrease the mobility and the longitudinal conductivity. As a result, the extrinsic AHE would diminish. Second, the Kondo theory for skew scattering depends on the spin waves of local moments at finite temperature that lead to asymmetric scattering. Recently, this scenario was generalized to the case with spin chirality, accounting for the AHE in chiral magnets. Moreover, it has been also proposed to explain the appearance of giant AHE in kagome metals AV3Sb5. However, the nearly vanishing is the Fermi energy, is the impurity concentration and is the impurity potential strength [17]. The approximate concentration of impurities to be the order of 1%, which includes the nonmagnetic impurities and paramagnetic impurities.

analysis of previous results--Let us turn to discuss the recent experimental results of AHE
in the kagome family compounds AV3Sb5. In Ref. [14] for the AHE in KV3Sb5, the AHE was attributed to the scattering of Dirac quasiparticles by chiral spin clusters of V atoms. The authors proposed a quadratic scaling between anomalous Hall conductivity and the longitudinal one, which is distinct from the linear scaling law for the extrinsic skew scattering mechanism for AHE. However, recent spin-muon scattering did find no evidence of sufficiently large local moment of V atom, thus the mechanism was ruled out. Moreover, Ref. [14] neither identified the existence of CDW order nor established the connection between CDW and AHE. In Ref.
[2], similar AHE was observed in CsV3Sb5 and was proposed to obey the quadratic scaling law as KV3Sb5 in Ref. [14]. The evolution of AHE under high pressure further revealed the possible connection between AHE and CDW order. The authors instead suggest that the giant AHE may come from enhanced skew scattering in the CDW state and large Berry curvature due to the kagome lattice . Recent ARPES experiments revealed the multiple energy scales for CDW order:   70 meV for a band forming a saddle point near the M points and the smaller gap (~18 meV) for a band forming Dirac cones [19]. However, the transport experiment in Ref. [2] did not unambiguously determine which energy bands for CDW phase would dominate the giant AHE.
That is, whether the Dirac fermions, the heavy fermions near the saddle points play a crucial role in the AHE? Meanwhile, the authors did not clarify the role of the electron bands near the G-point in the AHE. More importantly, the authors did not determine which scenario would dominate the skew scattering contribution to giant AHE, which is crucial to reach a microscopic model to quantitatively understand the giant AHE. Due to limitation of transport data from samples with fixed carrier density, there is no definitive magnitude of intrinsic AHE entirely from the Berry curvature of energy bands. In addition, some controversy between the wellknown linear scaling law for anomalous Hall conductivity and the longitudinal one [17] and the quadratic scaling law proposed in Refs. [2] and [14] need to be clarified.

our results--First-principles calculations and spectroscopy experiments reveal three distinct
Fermi surface pockets in all three kagome compounds AV3Sb5: (i) a circular pocket around the G-point that originates from the electronic states in the pz orbital of Sb (ii) Dirac cones with dominant , 2 − 2 and z 2 character (iii) heavy-hole bands of and orbitals of V atoms near the saddle points [20].

electrons near the G point--We first briefly consider the impacts of the electron bands near
the G-point. These topologically trivial bands do not support Berry curvature and thus not contribute to the intrinsic AHE. Moreover, none of these energy bands near the G-point have contribution to the extrinsic AHE primarily due to their irrelevance to the CDW transition.  [23][24][25] in the main text) implies that the real intrinsic AHE from Dirac bands is negligible. Recent numerical study shows that the extrinsic AHE is nearly independent of the longitudinal one and significantly smaller than the intrinsic one in the doped Weyl semimetals or Weyl metals [21]. It is inconsistent with the fact that the extrinsic AHE dominates over the intrinsic one in AV3Sb5. As a result, the total AHE including intrinsic and extrinsic parts from Dirac bands could not mainly account for the giant AHE in the kagome family compounds. It is worth noting that the particle-hole symmetry of the Dirac type quasiparticles implies a symmetric magnitude of AHE in both the hole and electron pockets. However, the significant difference in the evolution (both magnitude and trend) of the AHE in both electron and hole pockets in our experiments suggests that Dirac type quasiparticles with particle-hole symmetry unlikely leads to the giant AHE. Therefore, the giant AHE was mainly contributed by the heavy-hole bands with nonzero Berry curvature near the saddle points in the Brillouin zone.  [17] and 2D massive Dirac fermions [18], in which the extrinsic skew scattering can be dominant in AHE in the high conducting regime. Thus, our experimental findings establish a global picture of the AHE against the Fermi energy and further unveil its correlation with the possible chiral charge order in the topological kagome lattice AV3Sb5.

9-2. Side jump.
In this section, we would like to briefly clarify the side jump contribution to AHE, which is inevitable due to the disorder scattering. The side-jump mechanism could be viewed as a consequence of the anomalous velocity mechanism due to the Berry curvature acting while a quasiparticle was under the influence of the electric field due to an impurity [16]. The ultrahigh longitudinal conductivity in AV3Sb5 indicates the low concentration of impurities, implying that the side jump contribution is probably smaller than the intrinsic one. According to the empirical scaling law between the anomalous Hall conductivity and the longitudinal one [17], the side jump contribution is usually independent of the scattering time contributed from the intrinsic Berry phase. As shown in Figure 4a in the main text, the contribution that is independent of the longitudinal conductivity is about 1100 Ω −1 cm −1 , only 7% of the giant AHE observed in AV3Sb5. Thus, we conclude that the side jump mechanism made a minor contribution to the giant AHE. Note that these two contributions vary distinctly with respect to the material-dependent parameters such as the Fermi level, the exchange splitting and the strength of spin-orbital coupling.